Yes, SSDs can fail altogether on such an occurance, or display corruption on data you've not even touched (i.e. not written to; but still it is corrupted).
You would need an SSD of the third generation with a 'supercapacitor' to stop corrupting on power failure / abrupt disconnections. Usually the data that goes on SSDs is not very important and can be re-created (your OS drive should hold only system files). If you can map your my documents to another storage device that would be most ideal so you don't need anything on the SSD.
Corruption is still an issue with SSDs; the Supercapacitor third generation SSDs would finally be able to write safely, without fearing that it would corrupt its own storage when the power is lost.